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Laser Doppler Flowometry A.K.Khachatryan. Final Report, Phys
173, UCSD Laboratory of Optics and Biophysics. Performed under the
supervision of Dr. Phlbert Tsai Doppler Effect analysis has proven
to be a useful forensic tool for a variety of studies in physical
and biological sciences. Having been used in fields varying such as
astronomy, acoustics, vibrational analysis and flow research, it is
now a major contributor to the advancement of research in
biophysics and biomedical sciences. The objective of this paper is
to provide a report of a technique known as Laser Doppler
Flowometry (LDF) which uses relativistic doppler shift analysis to
determine the velocity of particles suspended in a moving fluid. A
direct application of LDF in medicine is in the field of
hemodynamics research that seeks to quantify the flow of blood in
human tissue. LDF is a highly favorable technique as it is
noninvasive and is performed without causing a disturbance in the
surrounding patterns of fluid flow or the tissue. Introduction and
Background Theory Relativistic Doppler Effect is a phenomenon
related to electromagnetic waves, in this case - light. It is the
change in frequency, and therefore the wavelength of light caused
by relative motion of the incident light beam and the moving
particle. As the two beams come in to contact with the particle at
their focal point, they become scattered and pick up the particle’s
velocity component, therefore acquiring a change in their
frequency. The relativistic doppler shift (RDS) formula reads
F!"##$%& =
2!"#(!/!)!!"
V! *
where F!"##$%& is the difference between the
frequencies of the two scattered beams, θ is the angle between the
incident beams, λ is the wavelength of the incident beams and
V! is the velocity of the particle.
Fig. 1. Beam scattering at contact with moving particle
fsb1 = fib1 + fib1(vpuib )c
= fib1(1+vpuib1λib
)
fsb2 = fib2 + fib2(vpuib )c
= fib2 (1+vpuib1λib
)
fdoppler = fsb2 − fsb1
fdoppler = 2sin(θ / 2)
λibvp
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Two laser beams are focused at a point where the scattering
particle is in motion. Fig. 1 shows IB1 ad IB2 (the 1st and 2nd
incident beams respectively) coming to a focal point where the
particle is in motion with velocity vector vp . Vectors
uib1 and uib2 are unit
vectors in the direction of the IB1 and IB2 respectively.
Methods Experimental Set-up To produce the desired scattering
effects for the RDS to work, an experimental set-up was designed as
shown in Fig. 2.
Fig. 2. Experimental Setup: Two-Beam Configuration for LDF A
helium-neon laser of power 8mW was used which produced a red beam
of light of 635nm wavelength. The beam was sent through a prism
that split the beam into two beams (Beam 1 and Beam 2 in Fig. 2)
perpendicular to one another. A mirror was used to collimate the
two beams. The parallel beams were then sent through an objective
lens that focused them at the center of the tube containing the
moving fluid with particles in it. A couple of other lenses were
then used to collect, collimate and lastly to focus the scattered
beams onto a photodetector. Flow Tube A gravity pump was made to
allow for fluid to pass through the tube at constant velocity. The
tube used was of 0.2x2.0 mm in dimensions and was much smaller in
size compared to the reservoir of 50ml volume. The relatively large
size of the reservoir is desired so that the change in pressure
throughout the reservoir is negligible relative to the flow tube in
the small amount of time period within which the data was recorded.
This allows for the assumed constancy of the velocity of fluid in
the flow tube.
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Data Analysis Initially, two laser beams are brought to a focal
point located at the position at which the speed of the moving
particle is to be taken. At this point the two beams interfere and
produce an interference pattern. The particle, moving through this
point with constant velocity and being of the same order of
magnitude in size as the distance between the interference fringes,
will scatter the beams in all directions. The scattered beams pick
up a velocity component form the scattering particle and change
their frequency. However, this change is so small compared to the
significantly larger original frequency of the incoming beams that
it is not realistically detectable. Therefore, instead of measuring
the change in frequency of individual beams, we look into the beat
frequency instead that results from the superposition of the two
beams as shown in Fig. 3.
Fig. 3. Interference pattern at the focal point of incident
laser beams and a graphical representation of the beat frequency of
the superimposed scattered waves. It is the superimposed doppler
shifter beams focused at the center of the photodiode that produce
a signal in the form of intensity of the light. The photodetector
is connected to a current to voltage converter. The signal is then
amplified by a factor of 104
so that the signal may be detectable. An oscilloscope was used
for live monitoring of the experiment. The signal in the form of
voltage vs. time is then recorded in MatLab. The application of
Fast Fourier Analysis allows to read the data in a frequency domain
where the observed peaks represent the beats. Results and
Discussion The experiment was conducted with a mixture of water and
polystyrene microsphere beads of 15 micron and 2 micron sizes and
with milk acting as a homogenous solution. It was expected that the
larger heavier beads would generate a more noticeable signal. The
expected doppler shift was calculated theoretically before data was
recorded to determine the consistency between the experimental and
theoretical results. The measurements were taken at various
velocities ** of the moving beads in the fluid.
** - See Appendix 1 for experimental measurement of bead
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Noise A test run was conducted with a stationary fluid to
determine and compare the noise to the velocity dependent signal of
a moving bead. The graphs below provide a comparison between the
noise and the signal. The axes in both graphs are on the same
scale. In the case of noise no evident peak is observed unlike in
the case of signal. This is to confirm that the experimental
measurements are indeed capturing a velocity dependent signal when
the fluid is in motion.
Signal Large Beads: Fig. 6 and 7 show measured peaks of 15
micron bead fluid in motion at two different velocities. In order
to calculate the theoretical frequency at which the peak should
occur, we experimentally measured the velocity of the fluid **
prior to analyzing the signal.
Fig. 4. Noise Fig. 5. Signal
Fig.6. Moving fluid – 15 micron beads: velocity of 3.125cm/s
with Fd(theoretical) - 23 kHz Fd(measured) ~ 3.5kHz
Fig. 7. Moving Fluid – 15 micron beads: Velocity - 5.75cm/s
Fd(theoretical) – 23kHz Fd(measured) ~ 4kHz
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The results showed a velocity dependent signal: the signal for a
bead moving at larger velocity produced a higher peak compared to a
slower moving bead therefore confirming the assumption that the
power of the doppler shifted frequency spectrum is directly
proportional to the velocity of the scattering particle. We also
see a small shift in the peak towards a higher frequency for the
particle with larger velocity also validating the presence of the
detected beats. However, there is an inconsistency between
experimental and theoretical frequency at which the peak should
appear. In both cases at the given velocities of the particle, the
peak was observed at a much smaller frequency than expected.
Factors contributing to inconsistent results might be due to
inaccurate experimental velocity measurements.
First, the 15 micron beads are large and heavy enough in size to
interfere with the assumed experimental variables. The
gravitational effects are higher on larger sized beads. This can
prevent the beads being uniformly mixed in the water and moving at
the same speed. Also assuming laminar flow, our experimental
measurements of velocity of the fluid could have been an inaccurate
representation of the actual velocity of the beads because the
large heavy beads are expected to move slower than the water around
them. Particles moving at various speeds in a non-uniformly mixed
fluid will scatter beams with different intensities causing a wider
range of frequency spectrum for the peaks. To correct for this
error smaller sized beads should be used.
Second, Boundary conditions were not taken into consideration. A
flow of a fluid in a tube is non-uniform in velocity throughout the
tube. Closer to the walls of the tube the fluid will move slower
than the fluid at the center of the tube. It is of utmost
importance to make sure the focus of the beams is placed at the
center of the tube to get a better estimate of the overall velocity
of the fluid. This can also be improved by using a larger tube to
better differentiate between the center and the boundaries of the
tube. Small beads and uniform homogeneous mixture: Same methods as
described for the 15 micron beads were used to collect data for the
2 micron beads as well as for heavily diluted milk. Milk was used
as an analogue to a homogeneous solution under the assumption that
all particles in it should move at roughly the same speed. The
results for the 2 micron beads very closely resembled the results
for milk, therefore, we only include data for milk sample.
A small peak was detected compared to much larger amounts of
noise for the sample. This makes it more difficult to interpret the
results. The most probable cause for this may be due to the size of
the beads or the particles in the milk being small enough for
Brownian motion to become relevant. The slower peak in the noise
for the stationary fluid is near the predicted velocity for the
Brownian motion ***. This makes it difficult to distinguish between
the true velocity of the particles moving in the fluid and the
velocity of the particles due thermal fluctuations. Therefore, the
experiment can be improved to produce better interpretable data if
the size of the particles is large enough so that the effects of
Brownian motion are negligible.
*** - See Appendix 2 for calculations of Brownian velocity of
beads
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Careful considerations were given to the boundary conditions in
the sample with milk. Data was collected for the beams converging
at i) the focal point of the beams being in the center of the tube
and ii) near the edges of the tube. Fig. 8 shows a test run for the
noise with stationary milk. Fig. 9 shows signal with milk moving
near the edges of the boundaries of the tube. For these two cases
the plots look almost the same, implying that near the edges the
fluid acts close to stationary. Fig. 10 depicts signal for data
collected from the center of the tube where the velocity of
particles is presumed to be larger. The signal is more noticeable.
This implies the correctness of our previous assumption of the
importance of placement of the beam focal point in the center of
the tube. Any deviations of the beams form the center of the tube
will result in inaccurate data. Conclusion
Experimental results certainly show a velocity dependent signal
as predicted by RDS. The beat frequency is detected in samples in
which the fluid is in motion. The intensity of the beats is
observed to be larger for faster moving fluids. Considering these
major findings it can be said that the experiment confirms
qualitatively the hypothetical predictions of the RDS up to a
certain degree.
However, the deviation of the measured frequency at which the
peaks appear from theoretically expected frequency is still an
impending problem that needs to be solved. In order to achieve the
desired quantitative results, future steps must be taken towards
improving the experiment which will include i) middle sized beads
(between 2 and 15 micron) to account for the effect of Brownian
motion and the non-uniform distribution of bead velocity in the
fluid, a better method for experimentally measuring the velocity of
the particles by ii) precisely placing the focus of the beams in
the center of the tube and iii) a larger flow tube in order to
correctly predict expected peak frequencies.
Fig. 8. Noise – Stationary Milk Fig. 9. Signal – Moving milk
near edges of the tube
Fig.10. Signal – Moving milk in the center of the tube
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Appendix 1 Experimental Measurement of Velocity of the Fluid
Initially the velocity of the fluid was calculated by timing the
fluid flow and counting the number of drops. This measured how much
volume of the fluid would flow through the tube in a given time.
The volumetric flow rate formula reads
Q = dVdt = Av where Q is the volume flow rate, A is the cross
sectional area through which the fluid flows and v is the velocity
of the fluid. However, calculation of velocity in this way might
lead to inaccurate results because the flow of water is not
continuous and uniform as drops flow out of the tube in a discrete
manner. This error was corrected for in the smaller bead and milk
samples by attaching an elastic tube to the flow tube and allowing
its end come into contact with liquid. This allows for continuous
uniform flow. The same volumetric flow rate formula was used in
this case also. Appendix 2 Calculation of Root Mean Square Velocity
of 2 Micron Beads Due to Brownian Motion Velocity of beads due to
Brownian is
where K is the Boltzmann constant, T is the temperature in
Kelvin and M is the molar mass of the beads. The physical
characteristics of the 2 micron beads to help calculate the molar
mass of the beads can be found Technical Data Sheet 238 titled
Polybead Plystirene Microspheres. Assuming room temperature, the
calculations resulted to be around 5.8x10-1 cm/s. This velocity was
then used in the RDS formula to calculate the frequency at which
the peak is expected which resulted in roughly around 4000Hz.
VRMS =3KTM
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References
Kalkert, C., and J. Kayser. Laser Doppler Flowometry. Rep.
UCSD/David Kleinfeld Laboratory.
Web. 10 Apr. 2014.
Keenan, L., and K. Chapin. Laser Doppler Velocimetry. Rep.
UCSD/David Kleinfeld
Laboratory. Web. 10 Apr. 2014.
"Polybead Polystyrene Microspheres." Polysciences, Inc Chemistry
Beyond the Ordinary. Web.
May-June 2014.
.
Tornos, J., M. A. Robelledo, and J. M. Alvarez. Laser Doppler
Velocimetry Experiment with a
Water Flow to Measure the Fourier Transform of the Time
-Interval Probability;
Comparison Between Experimental and Theoretical Results.
APS/Physical Review,
June-July 1989. Web. 2 May 2014.